Question

Two poles of equal heights are standing opposite to each other on either side of the road which is 90 m wide. From a point between them on the road, the angles of elevation of the top of the poles are 30\(^\circ\) and 60\(^\circ\) respectively. Find the height of the poles and the distances of the point from the poles. [Use \(\sqrt{3}\) = 1.732]

Hint:

The pole with the larger angle of elevation (\(60^\circ\)) will always be closer to the observation point than the pole with the smaller angle (\(30^\circ\)). This is a great conceptual sanity check!

Answer 1

Step 1: Understanding the Question:
We have two equal-height poles standing vertically on opposite sides of a 90 m wide road. From an intermediate point on the road, the angles of elevation to the tops of the poles are \(30^\circ\) and \(60^\circ\). We need to calculate the height of the poles and the exact location of the point.

Step 2: Key Formula or Approach:
Let the height of both poles be \(h\).
Let the distance of the point on the road from the pole with the \(60^\circ\) angle of elevation be \(x\) meters.
Then, the distance from the other pole is \((90 - x)\) meters.
We apply the tangent trigonometric ratio:
\[ \tan\theta = \frac{\text{Opposite}}{\text{Adjacent}} \]

Step 3: Detailed Explanation:

• For the first pole (with \(60^\circ\) elevation):
\[ \tan 60^\circ = \frac{h}{x} \implies \sqrt{3} = \frac{h}{x} \implies h = x\sqrt{3} \quad \text{(Equation 1)} \]

• For the second pole (with \(30^\circ\) elevation):
\[ \tan 30^\circ = \frac{h}{90 - x} \implies \frac{1}{\sqrt{3}} = \frac{h}{90 - x} \implies h = \frac{90 - x}{\sqrt{3}} \quad \text{(Equation 2)} \]

• Equate Equation 1 and Equation 2:
\[ x\sqrt{3} = \frac{90 - x}{\sqrt{3}} \]
Multiply both sides by \(\sqrt{3}\):
\[ 3x = 90 - x \]
\[ 4x = 90 \implies x = 22.5 \text{ m} \]

• Calculate the heights and distances:
- Distance of the point from the closer pole: \(22.5 \text{ m}\).
- Distance of the point from the farther pole: \(90 - 22.5 = 67.5 \text{ m}\).
- Height of the poles:
\[ h = x\sqrt{3} = 22.5 \times 1.732 = 38.97 \text{ m} \]


Step 4: Final Answer:
The height of the poles is 38.97 m, and the distances of the point from the poles are 22.5 m and 67.5 m.